Yes. Hiking in freezing rain is a common occurrence in the winter in the Southern Appalachians. Often the precipitation will change from rain to freezing rain to snow, and then back again as one climbs and descends a long hill. It's also common to have air temps in the high 20s and light rain, so it freezes all over your clothing and gear. It can be beautiful, but also dangerous.
I often find myself wearing too much clothing when this happens. Of course, it's hard to change once the freezing rain starts, so I try very hard to back off on the clothing selection in the morning -- just a lightweight merino wool base layer under my rain shell, and my rain pants, light w/b gaiters, and wool boxer briefs on the bottom. Everything is going to get soaked inside my rain gear, but the wool helps keep me warmish inside my shell layer as long as I keep moving.
And you have to keep moving, all day if necessary. Once you stop, it's critically important to get out of the rain, strip off all that soaking wet clothing, and put on dry, warm layers as quickly as possible. Walking all day under those conditions makes one a prime candidate for hypothermia - cold, wet, and exhausted.
I'm not as worried about using down insulation under these conditions, because I won't have those items out of the pack until I'm done for the day and inside my nice dry shelter. That's when having some thick microfleece base layers and a big puffy down jacket feels like heaven on Earth.